3.238 \(\int \frac{x^2}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=38 \[ -\frac{2 \left (b+2 c x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}} \]

[Out]

(-2*(b + 2*c*x^3))/(3*(b^2 - 4*a*c)*Sqrt[a + b*x^3 + c*x^6])

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Rubi [A]  time = 0.0593965, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{2 \left (b+2 c x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

(-2*(b + 2*c*x^3))/(3*(b^2 - 4*a*c)*Sqrt[a + b*x^3 + c*x^6])

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Rubi in Sympy [A]  time = 7.0245, size = 36, normalized size = 0.95 \[ - \frac{2 b + 4 c x^{3}}{3 \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(c*x**6+b*x**3+a)**(3/2),x)

[Out]

-(2*b + 4*c*x**3)/(3*(-4*a*c + b**2)*sqrt(a + b*x**3 + c*x**6))

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Mathematica [A]  time = 0.0269077, size = 38, normalized size = 1. \[ -\frac{2 \left (b+2 c x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a + b*x^3 + c*x^6)^(3/2),x]

[Out]

(-2*(b + 2*c*x^3))/(3*(b^2 - 4*a*c)*Sqrt[a + b*x^3 + c*x^6])

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Maple [A]  time = 0.009, size = 37, normalized size = 1. \[{\frac{4\,c{x}^{3}+2\,b}{12\,ac-3\,{b}^{2}}{\frac{1}{\sqrt{c{x}^{6}+b{x}^{3}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(c*x^6+b*x^3+a)^(3/2),x)

[Out]

2/3/(c*x^6+b*x^3+a)^(1/2)*(2*c*x^3+b)/(4*a*c-b^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(c*x^6 + b*x^3 + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.277464, size = 90, normalized size = 2.37 \[ -\frac{2 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c x^{3} + b\right )}}{3 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{6} +{\left (b^{3} - 4 \, a b c\right )} x^{3} + a b^{2} - 4 \, a^{2} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(c*x^6 + b*x^3 + a)^(3/2),x, algorithm="fricas")

[Out]

-2/3*sqrt(c*x^6 + b*x^3 + a)*(2*c*x^3 + b)/((b^2*c - 4*a*c^2)*x^6 + (b^3 - 4*a*b
*c)*x^3 + a*b^2 - 4*a^2*c)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(c*x**6+b*x**3+a)**(3/2),x)

[Out]

Integral(x**2/(a + b*x**3 + c*x**6)**(3/2), x)

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GIAC/XCAS [A]  time = 0.300747, size = 61, normalized size = 1.61 \[ -\frac{2 \,{\left (\frac{2 \, c x^{3}}{b^{2} - 4 \, a c} + \frac{b}{b^{2} - 4 \, a c}\right )}}{3 \, \sqrt{c x^{6} + b x^{3} + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(c*x^6 + b*x^3 + a)^(3/2),x, algorithm="giac")

[Out]

-2/3*(2*c*x^3/(b^2 - 4*a*c) + b/(b^2 - 4*a*c))/sqrt(c*x^6 + b*x^3 + a)